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Transcript of Chapter 6 Stability Criteria_nise
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Chapter 6: Stability1
Chapter 6Stability
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Chapter 6: Stability2
introduction
• C(t)=Cforced (t)+Cnatural (t)• For a bounded input, depend upon
natural response, the system can bestable, unstable or marginally stable• A system is stable if every bounded
input yields a bounded output ( ! ")• !f input is bounded but the totalresponse is not bounded, the system
is unstable
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Chapter 6: Stability3
Stability
• A system is stable if the naturalresponse approaches #ero as timeapproaches infinity$
• A system is marginally stable if thenatural response neither decays norgro%s, but remains constant oroscillates$
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Chapter 6: Stability4
&arginally stable
'ote that theroots are on theimaginary a is
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Chapter 6: Stability5
Figure 6.1
Closed-looppolesand response:a. stablesystem;b. unstablesystem
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Chapter 6: Stability6
Figure 6.2Common causeof problems infinding closed-loop poles:a. originalsystem;b. equivalent
system
(R-C)G=C C(1+G)=GR, C=[ G/(1+G )]R
(s)
(s)* + (s)-
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Chapter 6: Stability
Figure 6.3
Equivalentclosed-looptransferfunction
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Chapter 6: Stability!
Table 6.1Initial layout for Routh table
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Chapter 6: Stability"
Table 6.2Completed Routh table
!f no sign changes, stable
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Chapter 6: Stability1#
Figure 6.4
a. Feedbacksystem for E ample !"#;b. equivalentclosed-loopsystem
1#3#311#1###
1###)5)(3)(2(1###
)5)(3)(2(1###
1
)5)(3)(2(1###
)(1)(
23 +++=
++++=
++++
+++=
+
s s s s s s
s s s
s s s
sG
sG
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Chapter 6: Stability11
Table 6.3
Completed Routh table for E ample !"#
.%o sign changes, t%ounstable poles
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Chapter 6: Stability12
Table 6.4Completed Routh table for E ample !"$
356321#
)( 2345 +++++=
s s s s s sT
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Chapter 6: Stability13
Table 6.5%etermining signs in first column of aRouth table &ith 'ero as first element ina ro&
.%o sign changes/ t%o unstable poles
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Chapter 6: Stability14
Table 6.6Routh table forE ample !"(
123653)(
t$%$&'erte be%a&%riteria,1
356321#
)(
2345
2345
+++++=
=
+++++=
s s s s s s Dd
s
s s s s s sT
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Chapter 6: Stability15
Table 6.7
Routh table for E ample !")
56!4261#
)( 2345 +++++=
s s s s s sT
All#ero
0(s)=s1+2s3+4,05(s)=1s6+ 3s
0(s)05(s)
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Chapter 6: Stability16
C$&tr$l Sy te*16
*he au iliary polynomial
• *he equation that immediately precedesthe 'ero entry in the Routh array"
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Chapter 6: Stability1
Figure 6.5Root positionsto generate evenpolynomials:
A +B +C +or anycombination
!f t%o + realpoles, there mustbe t%o 7 realroots, other%iset%o roots are on
89a is
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Chapter 6: Stability1!
Table 6.8 Routh table for E ample !",
2#3!4!5"3"22122#
)( 23456! ++++++++=
s s s s s s s s sT
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Chapter 6: Stability2#
Figure 6.6Feedbackcontrol systemfor E ample !"!
2##61162##
)6116(2##
1)6116(
2##
)(1)()(
234
23
23
++++=
++++
+++=+=
s s s s
s s s s
s s s s sG
sG sT
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Chapter 6: Stability21
Table 6.10
Routh table for E ample !"!
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Chapter 6: Stability22
Figure 6.7Feedback controlsystem for E ample !".
123232
1
)23232(1
1
)23232(1
)(1)(
)(
2345
234
234
+++++=
+++++
++++=+
=
s s s s s
s s s s s
s s s s s sG
sG sT
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Chapter 6: Stability23
Table 6.11 Routh table for E ample !".
.%o poles in rhs/unstable
.hree poles in lhs
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Chapter 6: Stability24
Table 6.12 /lternative Routh table for E ample !".
23232it Rearra&+e
1232321
)(
2345
2345
+++++
+++++=
s s s s s
s s s s s sT
.he sameresult/ .%opoles in rhs/.hree poles in
lhs
0ut s= *d/ s and d
sho%s the samesign
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Chapter 6: Stability25
Figure 6.8 Feedback control system for E ample !"0
12!1"212!"64!241#31
)1"212!"64!241#3(12!1
)1"212!"64!241#3(12!
)(1
)()(
23456!
23456
23456
++++++++=
++++++++
+++++++=+
=
s s s s s s s s
s s s s s s s s
s s s s s s s s
sG
sG sT
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Chapter 6: Stability26
Table 6.13
Routh table for E ample !"0
( )
0(s)=s2+4s1+63s3+21/ t%o rhs (t%o sign changes), sot%o lhs, the remaining t%o on 89a is
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Chapter 6: Stability2
Table 6.14ummary of pole locations for E ample !"0
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Chapter 6: Stability2!
Figure 6.9Jason is anunder&ater+ remote-controlled vehicle thathas been used toe plore the &reckage ofthe Lusitania " *hemanipulator andcamera comprisesome of the vehicle1scontrol systems"
Courtesy of Woods ole !"ea#ogra$%i" stitutio#.
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Chapter 6: Stability2"
Figure 6.10
Feedback controlsystem for E ample!"2
K s s s
K
s s s
K
s s s
K
sG
sG sT
+++=
+++
++=
+=
1!
)11)((1
)11)(()(1
)()(
23
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Chapter 6: Stability3#
Table 6.15
Routh table for E ample !"2
!f no sign changeat the first column/
unstable!f there is a #eroro%, 8% is possible Stable
:;
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Chapter 6: Stability31
Table 6.16
Routh table for E ample !"2 &ith K 3 #(0!
0(s)= 4s3+ 64205(s)=62s
As there is no
sign changebelo%, there aret%o 8% poles
&arginally stable
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Chapter 6: Stability32
Figure 6.11*he F/45CRobot 6- )77can beconfigured for )-or ,-a is ofmotion" It is
seen heremoving andstacking bo es"
Courtesy of Fa#u" 'oboti"s.
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Chapter 6: Stability33
Table 6.17Routh table for E ample !"#7
s1
+6s6
+6:s3
+3::
( )
( )= 2+1# s1+6s6+6:s 3+3::=( 2+1#)
( 2+3y+2#)
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Chapter 6: Stability34
Stability in State Space
• System poles are e ual to the eigenvalues of the system matri A
• A =λ• (λ !9A) =:• = (λ !9A)9, :
• = ad8(λ !9A)
*det ( λ !9A)-:• det ( λ !9A)=:• >se det (s!9A)=:
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Chapter 6: Stability35
[ ]
526)et(
251#
1!2
13
251#
1!2
13#
##
##
##
)(
##1
#
#
1#
251#
1!2
13#
23 −−−=−
+−−−−−
=
−−−−=−
=
+
−−−
=
s s s A sI
s
s
s
s
s
s
A sI
x y
u x x
s(s94)(s+3)+6:+ :+ :(s94)+?s92(s+3)
=s692s39@s9?3
36
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Chapter 6: Stability36
Table 6.18Routh table for E ample !"##
&e rh , t.$ lh 0& table
3
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Chapter 6: Stability3
Table 6.19
Routh table for antenna controlcase study
3!
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Chapter 6: Stability3!
Table 6.20Routh table for 5F case study
3"
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Chapter 6: Stability3"
Figure (6.1
4#
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Chapter 6: Stability4#
Figure (6.2
41
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Chapter 6: Stability41
Figure (6.3
42
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Chapter 6: Stability42
Figure (6.4
43
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Chapter 6: Stability43
Figure (6.5
44
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Chapter 6: Stability44
Figure (6.6
45
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Chapter 6: Stability45
Figure (6.7
h b l46
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Chapter 6: Stability46
Figure (6.8
Ch 6 S bili4
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Chapter 6: Stability4
Figure (6.9
Ch 6 S bili4!
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Chapter 6: Stability4!
Figure (6.10Closed-loopsystem&ith pole plot
Ch t 6 St bilit4"
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Chapter 6: Stability4
Figure (6.11
/ircraft pitch loopmodel
Ch t 6 St bilit5#
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Chapter 6: Stability5#
Figure (6.128lock diagram of achemical process-controlsystem
Ch t 6 St bilit51
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Chapter 6: Stability51
Figure (6.13a. Soft Arm used for feeding;b. simplified
blockdiagram
Courtesy of )a*u%i+o )a,a-ura /a#derbilt #i ersity.
Chapter 6: Stability52
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Chapter 6: Stability
Figure (6.14
*o&ed vehicle rollcontrol
Chapter 6: Stability53
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Chapter 6: Stability
Figure (6.15Cutting force controlsystem
1997 .
Chapter 6: Stability54
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Chapter 6: Stability
Figure (6.16a. / magneticlevitationtransportationsystem;b. simplifiedblock
diagram 9 #220IEEE
a$a# ir i#es (%oto 'esear"%ers.
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Chapter 6: Stability56
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Chapter 6: Stability
Table 6.2Completed Routh table
Chapter 6: Stability5
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Chapter 6: Stability
Table 6.3Completed Routh table for E ample !"#
Chapter 6: Stability5!
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Chapter 6: Stability
Table 6.4Completed Routh table for E ample !"$
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Chapter 6: Stability61
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Chapter 6: Stability
Table 6.7
Routh table for E ample !")
Chapter 6: Stability62
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Chapter 6: Stability
Table 6.8Routh table for E ample !",
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Chapter 6: Stability65
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p y
Table 6.11 Routh table for E ample !".
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Chapter 6: Stability6
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p y
Table 6.13
Routh table for E ample !"0
Chapter 6: Stability6!
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Table 6.14ummary of pole locations for E ample !"0
Chapter 6: Stability6"
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Table 6.15Routh table forE ample !"2
Chapter 6: Stability#
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Table 6.16Routh table for E ample !"2 &ith K 3 #(0!
Chapter 6: Stability1
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Table 6.17
Routh table for E ample !"#7
Chapter 6: Stability2
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Table 6.18Routh table for E ample !"##
Chapter 6: Stability3
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Table 6.19Routh table for antenna controlcase study
Chapter 6: Stability4
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Table 6.20Routh table for 5F case study